Question

Explain the relationships among angle measure in degrees , angle measure in radians, and arc length.

Answer 1

Answer on Question #57973 – Math – Trigonometry

Question

Explain the relationships among angle measure in degrees, angle measure in radians, and arc length.

Solution

For angle $d{}^{\circ}m's''$ with $d$ integer degrees $m$ minutes and $s$ seconds the decimal degrees is equal to

$$d{}^{\circ} + \left(\frac{m}{60}\right){}^{\circ} + \left(\frac{s}{3600}\right){}^{\circ}$$

Angle measure of $x{}^{\circ}$ corresponds to angle measure of $x{}^{\circ} \frac{\pi}{180{}^{\circ}} = \frac{x\pi}{180}$ (in radians).

Angle measure of $y$ radians corresponds to $y \cdot \frac{180{}^{\circ}}{\pi}$ (in degrees).

In the following lines, $s$ represents the length of an arc of the circle, $\theta$ is the angle which the arc subtends at the centre of the circle, $r$ is the radius of a circle.

If $\theta$ is in radians, then arc length is $s = r\theta$.

If $\theta$ is in degrees, then arc length is $s = \frac{\pi r\theta{}^{\circ}}{180{}^{\circ}} = \frac{\pi r\theta}{180}$.

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